A Study of the Probability Distribution of the Balls in Bins Process with Power Law Feedback

TL;DR

This paper investigates the probability distribution in the balls in bins process with power law feedback, providing numerical and simulation evidence that the distribution follows a power law, extending previous results to multiple bins.

Contribution

It offers a recursive solution to the master equation and extends the understanding of power law tails in the process to multiple bins, supported by numerical and simulation evidence.

Findings

Probability mass function asymptotically follows a power law for b3>1

Tail of losers scales as b3-1 for multiple bins

Process produces power law distributions common in real-world phenomena

Abstract

We analyse the balls in bins process with feedback with primary focus on the power law feedback function $f(\omega)=\eta \omega^{\gamma}\,$, $\eta>0\,$ $\gamma \geq0\,$. Using the recursive solution to the master equation we find for power law feedback numerical evidence that the probability mass function for finite time scales asymptotically with $\omega^{-\gamma}\,$ for $\gamma>1\,$. We also provide simulations supporting a previous result by Oliveira (corollary to Theorem 4 in \cite{oliveira2009onset}) that the tail of the losers scale as $\omega^{-(\gamma-1)}\,$ but extending to $N \geq 2$ bins. We thus find evidence that the balls in bins process with power law feedback produces power law distributions as is common to many real world phenomena.